    # Concept Of Risk And Measurement Techniques Illustration 107

Suppose an investment in offshore oil exploration, there are two feasible results, the success of the project capitulating a payoff of \$45 per share with a possibility 0.35 and the failure capitulating a payoff \$5 per share with a possibility of 0.65. What is the anticipated value of investment per share.

Also determine what would be the investment per share in case of 5 probabilities as given below.

 Cash Flows Value in Million \$ - Xi Possibilities Pi 25 0.4 35 0.5 45 0.6 55 0.5 65 0.4

Solution

1. The anticipated value of investment per share

Formula to ascertain the anticipated value per share in case of two possibilities is,
__
E (X) or X =          P1X1 + P2X2

=          0.35 * 45 + 0.65 * 5

=          15.75   +          3.25

=          19

1. Formula to ascertain 2 probable outcomes is

E (X) or X =          P1X1 + P2X2 + …. PnXn
n
=          Σ          PiXi
t=1

Therefore,

=          (0.4 * 25) + (0.5 * 35) + (0.6 * 45) + (0.5 * 55) + (0.4 * 65)

=          10        +          17.5     +          27        +          27.5     +          26

=          108

Or

 Cash Flows Value in Million \$ - Xi Possibilities Pi Anticipated Value PiXi 25 0.4 10 35 0.5 17.5 45 0.6 27 55 0.5 27.5 65 0.4 26 Σ PiXi = 108

__
E(X) or X        =          108

Illustration 108

Determine the standard deviation for the following probable outcomes.

 Cash Flow Probability 15 0.15 25 0.25 35 0.35 45 0.45 55 0.55

Solution

 Cash Flow Outcome - X Probability Pi _ X – X _ (X – X)^ 2 _ (X – X)^2 Pi 15 0.15 - 20 400 60 25 0.25 - 10 100 25 35 0.35 0 0 0 45 0.45 10 100 4.5 55 0.55 20 400 11 _  X = 35 0 100.5

Σ = √ 100.5     =          10.02

Illustration 109

Determine the standard deviation and Coefficient Variation for the following.

Project A

 Cash Flow Probability 50 0.50 60 0.75 70 0.90 80 0.75 90 0.50

Project B

 Cash Flow Probability 40 0.75 55 0.85 70 0.95 85 0.85 100 0.75
1. Standard Deviation and Variance for the Project A is as follows:
 Cash Flow Probability _ X – X _ (X – X)^ 2 _ (X – X)^2 Pi 50 0.50 - 20 400 200 60 0.75 - 10 100 75 70 0.90 0 0 0 80 0.75 10 100 75 90 0.50 20 400 200 _ X = 70 550

n                      __
Standard Deviation σ       =          Σ          √ (X1 – X1)^ 2. P1
t=1

=          √550                =          23.45

Co-efficient Variance       =          VA                   =          σ A      =          23.45
R A                   550

=          0.04

1. Standard Deviation and Variance for the Project B is as follows:
 Cash Flow Probability _ X – X _ (X – X)^ 2 _ (X – X)^2 Pi 40 0.75 - 30 900 675 55 0.85 - 15 225 191.25 70 0.95 0 0 0 85 0.85 15 225 191.25 100 0.75 30 900 675 _ X = 70 1766.25

n                      __
Standard Deviation σ       =          Σ          √ (X1 – X1)^ 2. P1
t=1

=          √1766.25         =          42.02

Co-efficient Variance       =          VA                   =          σ A      =          42.02
R A                 1766.25

=          0.02

Therefore, the standard deviation as well as the variance is greater for Project B. This entails the relative risk of project B is comparatively lesser than that of project A.

Thus, as the anticipated value of project B is greater and its relative risk too is greater, the manager will decide to invest in this project.

Illustration 110

The table presents the probability allocation and anticipated value on investments. Determine the less risk project among the two.

 State of Nature Probability Pi Outcome Monetary Return Xi Investment in Project C Inflation Null 0.25 100 Reasonable Inflation 0.40 175 Huge Inflation 0.30 300 Investment in Project D Inflation Null 0.25 150 Reasonable Inflation 0.40 200 Huge Inflation 0.30 250

Investment Project C

Anticipated Value E(X)    =          P1X1 + P2X2 + P3X3

=          0.25 * 100 + 0.40 * 175 + 0.30 * 300

=          25        +          70        +          90

=          185

S D σ         =          √ P1 (X1 – E(X)) ^2 + P2 (X2 – E(X)) ^ 2 + P3 (X3 – E(X)) ^2

=          √ 0.25 (100 –185) ^2 + 0.40 (175–185) ^2 + 0.30 (300 –185) ^2

=          √ 0.25 (85) ^2 + 0.40 (-10) ^2 + 0.30 (115) ^2

=          √ 0.25 (7225) + 0.40 (100) + 0.30 (13325)

=          √ 1806.25 + 40 + 3997.5

=          √ 5843.75        =          76.44

Investment Project D

P1X1 + P2X2 + P3X3

=          0.25 * 150 + 0.40 * 200 + 0.30 * 250

=          37.5     +          80        +          75

=          192.5

S D σ         =          √ P1 (X1 – E(X)) ^2 + P2 (X2 – E(X)) ^ 2 + P3 (X3 – E(X)) ^2

=          √0.25(150–192.5)^2+0.40(200–192.5)^2 + 0.30 (250–192.5)^2

=          √ 0.25 (42.5) ^2 + 0.40 (-7.5) ^2 + 0.30 (57.5) ^2

=          √ 0.25 (1806.25) + 0.40 (56.25) + 0.30 (3306.25)

=          √ 451.56 + 22.5 + 991.875

=          √ 1466 =          38.28

1. It is unambiguous from the above that anticipated return from investment Project C and D are slightly different, that is 185 and 192.5.
1. However the investment Project C integrates double the risk as compared to investment D that is 76.44 and 38.28 correspondingly.
1. Thus, the manager would choose Project D which has lesser risk comparatively.
1. It is thus unambiguous that decision making involving risk is based not only on the anticipated value but also on the degree of risk incorporated and in the person’s approach toward risk.
1. It has to be noted that the scrutiny of decision making by a person integrating risk and uncertainty is helpful not only for choosing the investments but also scrutinising between any performance integrating risk and uncertainty.
1. Therefore, selection among different ways of accomplishment that vary in both anticipated value and risk, needs to be scrutinised the individuals preference towards risk.

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